# The linear stability of Reissner-Nordstr\"om spacetime for small charge

## TL;DR

This paper proves the linear stability of small-charged Reissner-Nordstr"om black holes against gravitational and electromagnetic perturbations, showing solutions remain bounded and decay, supporting the understanding of black hole stability.

## Contribution

It establishes the linear stability of Reissner-Nordstr"om black holes with small charge, including decay rates and gauge control, extending previous stability results.

## Key findings

- Solutions remain globally bounded outside the black hole.
- Perturbations decay polynomially over time.
- Decay rates match those expected in nonlinear stability.

## Abstract

In this paper, we prove the linear stability to gravitational and electromagnetic perturbations of the Reissner-Nordstr\"om family of charged black holes with small charge. Solutions to the linearized Einstein-Maxwell equations around a Reissner-Nordstr\"om solution arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearized Kerr-Newman metric. We express the perturbations in geodesic outgoing null foliations, also known as Bondi gauge. To obtain decay of the solution, one must add a residual pure gauge solution which is proved to be itself controlled from initial data. Our results rely on decay statements for the Teukolsky system of spin $\pm2$ and spin $\pm1$ satisfied by gauge-invariant null-decomposed curvature components, obtained in earlier works. These decays are then exploited to obtain polynomial decay for all the remaining components of curvature, electromagnetic tensor and Ricci coefficients. In particular, the obtained decay is optimal in the sense that it is the one which is expected to hold in the non-linear stability problem.